It is known that (REF ) yields the solvability of \(L^p\)  Dirichlet problem, with \(p\ge q^{\prime }:=\frac{q}{q-1}\) . See e.g. [1]} Theorem 1.7.3, or [2]} for this argument. Therefore, to prove the existence part of Theorem REF , it suffices to show the Carleson measure estimate (REF ). Indeed, we prove the following:
